The fix-the-wobbly-table theorem

The fix-the-wobbly-table theorem

Welcome to another Mathologer video.
Today’s video is about the absolutely
wonderful wobbly table theorem. Some of
you may already be familiar with a very
pretty special case of this theorem
which became quite well known a few
years ago when Numberphile dedicated a
nice video to it. This special case of
the theorem runs as follows: take a
square table like the one over there. At
the moment this table is hovering above the
floor – a bit unusual behavior for a table
but perhaps it’s due to the ghostly
presence at our seance. Anyway the feet of
our spooky table are blue and the points
directly below the four feet are marked
in green. The ghost then disappears and the
table plunks onto the surface. Chances
are that it will wobble, right? Not
because the ghost is still playing games
but just because the floor is uneven. How
do we stabilize the table? Well, usually
people will wedge a napkin or whatever
under one of the wobbling feet, like this.
However, the wobbly table theorem says
there’s another method. The theorem says
that if the floor is not too crazily
uneven, then you can stabilize the square
table simply by rotating the table on
the spot, like in this example. So just
rotate it and eventually all four feet
will be touching the ground. Super neat
isn’t it? And this actually works very
well in practice. Try it on your next
cafe trip for example. And this is only a
small piece of some nice and mostly
unknown table turning maths which also
applies to non square tables. Well let’s
see. Now before we get going I have to
make two very important disclaimers. First when I’m talking about a square
table I always mean that the feet of the
table form a square. Of course, that’s not
always true, even if the table surface is
square. Often enough real-life wobbling
is not caused by the uneven floor but by
the table having feet of unequal lengths.
For example, over there we’ve got a table
on a nice flat surface but one of the
legs has been gnawed at by
a beaver or a ghost or whatever. Anyway
one leg is shorter so even though the
table is standing on a flat ground
obviously there’s no way of shifting the
table around to make it stable. Well we
could turn it over with its legs in the
air but that would be cheating. So, again,
by a square table we mean the feet of
the table form a square. Second
disclaimer. By having stabilized a table
we do not mean that the tabletop is
necessarily horizontal. Usually the
stabilized table will still slope in
some manner consistent with the uneven
floor. Of course such a slope can also be
annoying but definitely not as annoying
as a wobble and typically in real life a
slight unevenness in the ground will
translate into a slight and not too
annoying slope of the tabletop. With
those two disclaimer stored away it’s
time to have some serious fun turning
the tables. Now to do that
we’ll start with Homer Simpson. Of course
it’s never the wrong time to introduce
some Simpsons into the discussion. Recall
that my last video was all about Fourier
series and how to draw complicated
pictures like this Homer drawing with
epicycles. I’ll announce the winner of
the epicycle competition at the end of
this video. For now let me reCYCLE 🙂
Homer as a warm-up exercise for our
table turning. Let’s begin by framing
Homer in a rectangle like this. I call
this sort of rectangle a hugging
rectangle because Homer touches all four
sides of the rectangle. Of course there are
infinitely many hugging rectangles, one
for each possible orientation. Here are a few
more. There’s one and there’s another one
and one more. Now the aspect ratios of
Homer’s hugging rectangles will vary but
it turns out that one of them is special.
One of the hugging rectangles is a
perfect square. And that was not luck. It
turns out that any picture will have a
hugging square. Not obvious at all but
it’s true. We’ll give the idea of a proof in
a minute but first here a couple more
examples to illustrate. Apple, Apple, Apple,
and here is its square. Linux penguin, and
here’s its square. Fish … square. Just a few
random points … square. Now it can happen
that the shape has more than one hugging
square. For example, all hugging
rectangles of circles are squares.
Question for you: Are there any other
shapes all of whose hugging rectangles
are squares?
Leave your answers in the comments.
Here’s a hint. Take a close look at the
Mathologer homepage. Okay here’s the gist
of a proof that every picture has a
hugging square. We start with any hugging
rectangle as in the example over there
and color pairs of opposite sides red
and blue. Now let’s rotate this rectangle
through an angle of 90 degrees so that
at every stage we have a hugging
rectangle. Alright the key observation
is that the dimensions of the hugging
rectangle will change slowly as we
rotate with no sudden jumps. Using maths
jargon we would say that the dimensions
change continuously as we rotate. Now
having rotated through 90 degrees we’re
back where we started, with the same
hugging rectangle. Let’s double check. Hmmm, yep exactly the same right? Well not
quite. Have a closer look. What has
happened is that the red and blue sides
have swapped places. That may not seem
important but it’s the key to capturing
our hugging square. Okay back to the
beginning zero degrees. Let’s record the
difference between the red and blue
lengths as we rotate. To begin red is
longer than blue, so the starting
difference will be what? Positive
negative or zero? Well positive, of course!
However, because of the swapping, at the
end of the 90 degree turn red will be
shorter than blue and so the difference
will be negative. Now since the hugging
rectangle changes continuously this implies that
the difference red minus blue will also
change continuously, … from positive to
negative. And this means that at some
point the difference has to be zero. But
the difference being zero means that red
is equal to blue, that the red sides are
just as long as the blue sides. And, of
course, that means that at this instant
we have a hugging square. And that works
for any picture whatsoever, not just
Homer. Super neat, isn’t it? Just start
with any hugging rectangle, start
rotating and you’re guaranteed to come
across the hugging square. What does all this have
to do with table turning? Well a very
similar above-then-below argument shows
why, as you turn your wobbling square
table through 90 degrees you can expect
to come across a stable position with
all four feet touching the ground. Okay
mathematical seatbelts on? Then here we
Say that square table is on the ground
and it’s wobbling with the two blue feet
touching the ground and the red feet
moving up and down as the table is
rocking back and forth. Now wobble the
table such that the two red feet are
exactly the same vertical distance from
the ground. We’ll call that an equal
hovering position. Now let’s rotate the
table 90 degrees,
always with the two feet in the air in
an equal hovering position. Here we go.
Okay, how have we ended up? Well the table
is exactly in its starting position,
except red and blue have swapped. Now
it’s the red feet on the ground with the
blue feet hovering. But obviously the
swapping is only possible at an instant
where both the blue feet and a red feet
are on the ground. We can also graph this
as we did for the hugging rectangles.
Let’s do it. Here red and blue stand for
the vertical distances of the red and
blue feet from the ground. Then just as
before this difference function starts
are positive and ends up negative
and so should be exactly zero at some
instant along the way. As earlier we
concluded both red and blue are the same
at this point. However, since at least one
of red and blue is zero at all times, at
this special pink zeroing time both the
red and blue distances must be zero. This
means that all four feet are touching
the ground and our table has been
stabilized. Fantastic! Okay, so a square
table can always be balanced by rotating
it. Always? Really? Well, let’s put it
like this: I’ve been using this trick for
decades to stabilize real-life tables
and it’s never failed me. On the other
hand, it’s quite easy to conjure up
scenarios in which different parts of
our argument break down and/or
stabilizing by rotating won’t work.
Second challenge for you today: Try to
come up with some setups that foil our
argument. Just to get things going here’s
something very simple. When we lower the
table over there its top will hit the
top of the mountain and the feet will
never reach the ground. n=Now that we have
the square table sorted give or take
some finicky details let me tell you a
little bit about the history of
mathematical table balancing, the broader
scope of the nifty unwobbling-by-turning argument and the part my friends
and I played in turning this intuitive
argument into a nailed down mathematical
theorem. Okay
hands up who knows this guy. Marty’s
hand is up. Well out there in YouTube
land not many people are nodding their
heads. Marty this is terrible, our hero
has been forgotten. This is Martin
Gardner, by far the greatest popularizer
of mathematics of all time. Starting in
the 1950s and for a quarter of a century
Gardner wrote the hugely influential
Mathematical Games column in Scientific
American. He’s also the author of more
than 100 books on popular mathematics,
magic tricks, and so on. Gardner inspired
more people of my generation to become
than anybody else. For example you can
only watch this video because I really
got into maths as a result of reading
Gardner’s books. And if you enjoy
Mathologer videos it’s mostly because of
the lessons I’ve learned from Martin
Gardner. He was a master at explaining
real mathematics as simply as possible,
and no simpler. In fact much of popular
mathematics in books or on YouTube or
wherever have their origin in a Martin
Gardner column. Hexaflexagons, Conway’s
Game of Life, Dragon fractals, public key
cryptography, wobbly tables. Yes table
turning. In the May 1973 issue of his
Mathematical Games column Gardner
challenged his readers to discover a
version of our heuristic argument for
why stabilizing a square table by
turning should work. He then supplied an
answer to his puzzle in his next column.
Gardner credits Miodrag Novakovic
and Ken Austin as his source for the
wobbly table trick. I actually tracked
down Ken Austin about 15 years ago and
he told me that in fact it was his
friend Miodrag who discovered the trick
and how to justify it around 1950. Ken
only communicated the trick to Martin
Gardner. Miodrag’s version of the table
turning is actually a bit different from
the one I showed you and is also well
worth checking out. Yet another version
of the argument is presented by the
prominent German mathematician Matthias
Kreck who is the star of Numberphile’s
2014 table turning video, also definitely
check out that one. Now for a surprising
twist have a look at the last sentence
of Martin Gardner’s write-up: “A similar
argument can be applied to wobbly
rectangular tables by giving them 180
degree rotations, Now of course that’s
very welcome news since out in the wild
most tables are rectangular but not
square in shape. And there are lots of
other objects whose feet form rectangles
to which all this applies. For example I
personally use stabilizing by rotating
quite often with my trusty stepladder.
Anyway why does everybody only go on
about square tables if this all works
much more generally for rectangular
tables. Well when you try to adapt this
slick argument for square tables to
general rectangular tables you’ll find
that this is not nearly as
straightforward as Gardener’s
off-the-cuff comment suggests. Now before
I show you how to extend the argument to
let me tell you about the almost
definitive paper on wobbly tables which
a couple of friends and I published in
2007, in the Mathematical Intelligencer.
There Marty is also taking a bow. This
paper was all about making the wobbly
table argument into a nailed down
mathematical theorem by hammering out on
which surfaces a perfect rectangular
table can be stabilized by turning it on
the spot. One of the main results of our
paper is that stabilizing by turning is
guaranteed to work if the ground and the
table have the following properties:
First the ground. The ground has to be
Lipschitz continuous with associated
angle of at most 35.26 degrees.
Whoa that sounds scary
but it’s not really that bad. All it
means is that the ground is given by a
continuous function and that between any
two points on the ground the ground
slopes at an angle of at most 35.26
degrees. Another way of putting this
is to say that when you slide around the
cony 3d counterpart of the green wedge,
all of the ground will always be below
the wedge. So the scary term Lipschitz
continuous amounts to a maximum slope
requirement for our ground. Okay so
ground continuous and not too steep, check!!!
And what about the table?
Well obviously in line with our
introductory disclaimer the feet should
form a rectangle and we also require a
minimum leg lengths. If the legs are at
least half as long as the diagonals of
the tabletop then nothing can go wrong
in the way of little hills bumping into
the tabletop. So what Bill, Marty, Reiner
and I proved in our paper is that as
long as the surface is continues and not
too sloppy and as long as the table legs
are sufficiently long a perfect
rectangular table can be stabilized by
turning. The heart of this theorem is the
extension of the nice heuristic argument
for square tables but to be able to use
this argument we had to do some pretty
hard work to make sure that our
conditions guarantee that the table
always rotates as nicely as suggested by
the animations that I showed you earlier.
The idea underlying the proof and many
similar proofs is the so-called
Intermediate Value Theorem. This is a
mathematically rigorous version of it
the intuitively obvious fact captured
by our positive-to-negative diagram, the
fact that if a continuous function
changes from positive to negative it
will be 0 somewhere along the way. In
fact, in the proof we use the
intermediate value theorem not just once
but about a zillion times and for those
of you keen on the gory details of the
tricky proof and some of the other nice
results in our paper I’ll provide a link
in the description. Now before we look at
the proof for rectangular tables, here is
an interesting open problem. Is there any
non-rectangular four-legged table which
can always be stabilized by turning. For
example, what about the half-hexagon
table here. So is it possible to
stabilize this or some other mutant
table by rotating? Let me just tell you
about one neat observation. It turns out
that it is important whether or not the
feet of a table lie on a circle. Well
that’s true for our half-hexagon table
and, of course, it’s also true for all
rectangular tables. Why is this circle
business important? Well, it turns out
that if the feet of a table are not on a
circle, then there is definitely a not
too sloppy surface on which this table
will always wobble. Last challenge for
you today, prove this non-concircular-
feet-implies-doomed-to-wobble theorem.
Okay now for the hardcore Mathologer
fans let me sketch
how our simple heuristic rotation
argument for stabilizing square tables
can be extended to rectangular tables.
Okay deep breath. As with square tables
we’ll rotate rectangular tables such that at
all times they are at an equal hovering
position. So at all times throughout the
rotation either the blue pair of feet is
on the ground with the red feet hovering
an equal distance above the ground, as
pictured over there, or vice versa. So if
we can show that at some instant during
our rotation we are guaranteed to have
the blue feet on the ground and there’s
some other instant where we’re sure that the red
feet are on the ground, then we can argue
with continuity as before that somewhere
in between all four feet have to be on
the ground. Now an obvious difference
between the square and rectangular cases
is that a 90 degree turn won’t work for
a general rectangular table. We have to
rotate 180 degrees to bring it back to
the starting position.
But now the problem is that after a 180
degree turn the same pair of feet are on
the ground as at the start. Let’s have a
look. There, turn, same feet on the ground.
So unlike in the square case the end
position after the half turn is not
distinguished in any way from the
starting position.
So our rotation in this case doesn’t
automatically provide us with that
second crucial position where the two
feet that were hovering at the start end
up on the ground. Instead, I’ll show you
that the crucial position always exists
using a proof by contradiction. So let’s
assume that the opposite of what we want
to show is true let’s assume that
somewhere in the universe there’s a
rectangular table and a surface on which
the table cannot be stabilized by
turning. We’re in Fantasyland now and so
to keep things as uncluttered as
possible I’ll only show you the table
and the underlying green points on the
ground. Now
if this table cannot be stabilized by
turning, then the blue feet
must stay on the ground throughout
the rotation and the red feet will be
hovering in the air all the time,
something like that. Alright, blue feet on the
ground and red feed hovering all the
time. Let me now show how this assumption
leads to a contradiction, which then
finishes the proof, modulo all the
finicky details contained in our
Intelligencer paper. Let’s highlight the
rectangle formed by the feet, and let’s
also highlight the z-axis around which
we’re rotating. That brown line is the z-axis. And let’s get rid of the
non-essential bits. Now we make sure that
the center of the rectangle will be on
the z-axis throughout the rotation, like
that. So the center just wanders up and
down the z-axis as we turn. Okay two
important observations. First, since we
are rotating through 180 degrees you’ve
just seen all possible balancing
positions of our table, with the center
on the z-axis. There are no other
balancing positions like this. Second,
let’s call the height of the center of
the table in a certain position the
elevation of the table in this position.
Now, as we rotate, there will be a
position of the table where the
elevation is at a minimum. Let’s see, okay.
so it’s wandering. Now where’s the minimum?
Here, okay, did you spot it? Well, there it
is. We’re almost ready for the punchline.
Notice that if the diagonal connecting
the red points is translated straight
down, the diagonal will connect the green
dots underneath. But this means that we
can rotate the table into a new balancing
position with the blue feet ending up at
those two green points. But this new
balancing position is absolutely
impossible. Why because the elevation of
the new balancing position is lower than
the supposedly minimal elevation that we
started with and that’s the
contradiction. Pretty tricky but also
pretty cool don’t you think?
And that’s it for today as far as table
turning is concerned. I’ve just got one
more thing to do before I sign off,
announce the winner of our epicycle
competition from last time. Lots of nice
ideas and implementations which I’ve
listed in my comment pinned to the top of
the epicycle video comment section. The
one I settled for as the winner is by
Tetrahedri who succeeded in doing
something extra special. Their epicycle
system simultaneously draws out three
iterations of a space-filling curve how
neat is that? Check out the details of
how they did it in the description of
their video. Okay Tetrahedri please send me
your contact details via a comment and
I’ll send you Marty’s and my book. And
thanks again to everybody who took part
in this competition. And that’s it for
today. Happy table-turning.

100 thoughts on “The fix-the-wobbly-table theorem”

  1. 19:56 "if this table cannot be stabilised by turning, then the blue feet must stay on the ground throughout the rotation and the red feet will be hovering in the air all the time" it's not obvious why that must be the case. its logical equivalent also says that if you turn the table that way that at at least one position the blue feet don't both touch the ground, it follows that the table can be stabilised which is obviously false

  2. Oh, dear. When I did this, I did it by the ratio of red side to blue side, going from above 1 to below 1. Does that mean I lose a point? (Disclaimer: I did have one professor who did take a point off because 0 was "different" from other numbers so we could use positive and negative but not above and below any other number.)

  3. easier with a round table right? Forget it with a square or rectangular table unless you are lucky, thin and have lots empty space around.

  4. a target, but that would be cheating since the only part the rectangle could hug is essentially a circle.
    squares themselves also have this property

  5. Omg this theorem is so useful. Whenever I’m holding a seance where the table has an uneven leg and is on sine waves, this helps so much?

  6. It should be possible to extend this theorem to any 4 legged table, irrespective of the layout of the legs.

    Provided that there isn’t an obstacle artificially supporting the table, then 2 opposite legs must be in constant contact with the floor. If rotating the table can cause the other pair of legs to be in constant contact, then the direction of wobble will change.

    As long as the floor doesn’t have any sudden level changes (steps), and a rotation can change the direction of wobble, then a rotation can eliminate the wobble irrespective of the individual paths of the legs during rotation.

  7. Here is a wobbly table cartoon I drew 12 years ago…

    Felix Sputnik

  8. I don't feel like this is a full argument, at least for the square table. I think a full argument would be, any 3 points can touch the ground. If you turn it, 3 new points can touch the ground. At some point in that transition, you can have all 4 points touching the ground

  9. Порадовало, что качающийся столик по-английски – "во бля тейбл"

  10. Мне всегда нравилась эта теорема, потому что она имеет важные практические применения в обыденной жизни. Например, я однажды применил её в магазине при покупке стола – он качался, и вращение не помогало его поставить ровно. Аргументируя этим, я заставил продавцов принести другой экземпляр со склада.

  11. Intuitively the square case should also have the leg length constraint. Also what exactly is the flaw in your proof that the maximum slope constraint becomes necessary?

  12. If the ground is not continuous or not differentiable, then there's a chance the rotation won't work.

    Because a noncontinuous or non differentiable floor means it's possible for the red/blue difference to be non continuous, and thus skip past 0.

    An example of a non differentiable floor would be the corner of a sheer cliff on the side of the Grand canyon.

  13. I wonder if it is possible to find out if all legs of the table are the same length?

    What if I rotate the table first 180 degrees and if it’s wobbling then in another direction as before, we should have defect legs?

  14. 0:00 Video Starts

    1:53 Disclamers

    1. Table legs need to bof equald lenghts

    2. Table is "stable" in the sense that all parts of the tableis touching the ground

    . In real life this is not a big deal as it is hardly noticeable so the application of this

    model to real life still holds good

    3:48 "Hugging rectangles"/ Intermediate value theorem

    6:10 Using calculus to find hugging rectangles of different shapes

    7:48 Simulation of the model for squares

    8:47 Using "Hugging rectangels" principle to find points of stability

    12:10 The different models of table turning , Miordrag's and Martiez Krieg "Bier gartne guy"

    14:28 The assumptions (Lipschitz continous and leg lenghts and stuff)

    16:15 Intermediate value theorem

    17:05 Possibility of other shapes of stablisation

    17:30 The reason why it needs to be on a circle

    18:00 The full showcase of the model

    19:07 Rectangular case

    19:34 Proof by contradiction

  15. can you explain the example at 10:00 also the 35.5 degree thing like lipchitz feels alright but 35.5 just seems like some random number

  16. That's all good and nice with the wobbly table, but you might find yourselves being seated a bit uncomfortable after you also level all those pesky wobbly chairs around it :o)

  17. Wouldn't a reuleaux triangle have a square as every hugging rectangle… AND it be the same, stationary square for every orientation of the shape…?

  18. Is there a refrigerator door theorem? How do I make my refrigerator slightly un-level in a way that always makes the door try to close by itself due to gravity, no matter what position it is open to? My freezer does this, but my refrigerator is opposite and gravity makes the door try to open more once it's open past a certain point. Instead of randomly adjusting all the feet, it would be more fun to figure out the math on how to make this happen. I also have other doors that I would prefer gravity to tend to open, and yet others I would prefer gravity to tend to close, and even others I would prefer to just stay where I put them no matter what position they are in. I think that this should be possible by adjusting the hinges, perhaps with a shim behind either the upper or lower hinges… but again, just randomly doing things is not as good as figuring out the math behind them.

  19. I figured this out on my own years ago while trying to orient my scale to have all four feet firmly on my non-level floor. I always found that I could eventually find a rotation where all four feet were on the floor.

  20. Any shape with rotational order of symmetry 2^n, n an integer greater than or equal to 2, will have all its hugging rectangles as hugging squares.

  21. For the rectangle case, you can also do a contraction along one axis of the table and the surface to reduce the rectangular case to the square case.

  22. I'm pretty sure you can create a disturbance in the continuous surface of the floor that prevents the table from finding a secure position

  23. This morning I was sitting at a wobbly table with my son, and we we're getting increasingly annoyed. I recalled this video, and told my son the wobbles might go away if we rotated the table. After a one eighth counterclockwise turn, the wobble decreased. Another quarter counterclockwise turn and the wobble was gone!

  24. Reuleaux triangle also has all hugging rectangles as squares….
    & all other many many types of fixed width shapes as well

  25. It looks to me as though you're using a vertical projection of the corners of the table to find the contact points. But as soon as you rotate the the surface of the table about some combination of the X and Y axes ("tilting" the table), your corner projection changes from its original aspect ratio to some other aspect ratio. (In the case of the square table, when you tilt the table, the projection goes from square to rectangle.)

    I don't know that this causes an insoluble problem for the intermediate value theorem, since all such projections would also be changing continuously, but I'm not sure that the way you've presented the proof here is sufficient to make your case.

  26. The table being able to rotate relies on the assumption that there exists an angle you can turn the table to, at any given position of the table, such that the 2 red legs are equally distant from the ground. Given a sufficiently large dip in the height of the ground, this is no longer true.

  27. Where does the 35.26° requirement arise from? Shouldn't it be sufficient for the surface to be continuous for the intermediate value theorem to apply?

  28. Thanks for reminding me of this! I once had an idea that this could be a way to get the moral equivalent of the Intermediate Value Theorem in an intuitionistic logical framework. I will think about this a bit harder and try to make the vague idea more concrete, ….

  29. For the hugging square puzzle, Rouleaux "polygons" (I don't know how else to call them) are another possibility. Of course, regular squares also have only hugging squares, as opposed to hugging rectangles.

  30. Do you have any idea how can I implement an animation in GeoGebra that rotates a rectangle, always touching the image, like the one you have in the video? (Always beeing a hugging rectangle)

  31. I think your going for the fact that any curve of constant width has a hugging square that you can rotate about it like a circle, which I learned from a different video of yours. Only a year and a bit late to make this comment!

  32. non circular feet wobbling counter example is trivial, simply have a circular groove on the leg outside the circle at a different height to the other legs, but constant. No matter how you rotate it the hight difference will always cause a wobble.

  33. All hugging rectangles of circles and squares are square, but also octogons and any regular poligon with 4*n+4 sides. For example a 12-gon.

    You want proof? Eat my shorts.

    EDIT: actually, any shape with 4-way symmetry.

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